Theory

The steady solvers couple pressure and velocity with the SIMPLE algorithm (Patankar and Spalding, 1972): each iteration solves the momentum equation with the current pressure, then a pressure equation that makes the velocity field conserve mass. The pressure correction of plain SIMPLE neglects the velocity corrections of the neighbouring cells, so it overshoots; only a fraction $\alpha_p$ of it is kept, equation (1). SIMPLEC (Van Doormaal and Raithby, 1984) keeps a consistent form of those neighbour terms (OpenFOAM: consistent yes) and needs no pressure relaxation, so $\alpha_p = 1$ .

The pressure is relaxed as a FIELD (OpenFOAM relaxationFactors.fields), whereas for velocity and the turbulence quantities the EQUATIONS are relaxed (see Velocity relaxation).

On auto all relaxation factors move together with the solver settings:

solver setting	p	U (start-up / final)	k, ω	ρ	e
incompressible, SIMPLEC (default)	1.0	0.7 / 0.9	0.7	-	-
incompressible, plain SIMPLE	0.3	0.7 / 0.7	0.7	-	-
compressible, SIMPLE (auto)	0.3	0.7 / 0.7	0.7	0.1	0.7
compressible, transonic SIMPLE	1.0	0.9 / 0.9	0.9	1.0	0.8
compressible, pseudo-transient	not relaxed (all factors 1)

For the compressible solver, auto uses plain SIMPLE (consistent no) whatever the SIMPLEC switch says.
A typed pressure relaxation makes the SIMPLEC switch take effect as set: with SIMPLEC on, the other auto factors then take the SIMPLEC values (U 0.7 / 0.9, k and ω 0.7, ρ 1.0, e 0.9 with heat transfer, else 0.7).
If the start-up phase diverges, the run restarts once from the initial fields with the conservative set p 0.3, U 0.7, k 0.7, ω 0.7, ρ 0.1 and consistent no. The pseudo-transient solver halves its Courant number instead.

Equations

p^{n+1} = p^{n} + \alpha_p\left(p^{*} - p^{n}\right)

(1)

\alpha_p + \alpha_U \le 1.1 \quad \text{(plain SIMPLE, a typed } \alpha_p\text{: otherwise a warning)}

(2)

where	meaning	unit
$p^{n}$	Pressure of the previous iteration	Pa (m²/s² incompressible)
$p^{*}$	Pressure after this iteration's pressure equation	Pa (m²/s² incompressible)
$\alpha_p$	Pressure relaxation factor	-
$\alpha_U$	Velocity relaxation factor (final phase)	-

Default and rule

Default

auto: 1.0 (SIMPLEC), 0.3 (plain SIMPLE and every compressible run except transonic), 1.0 (transonic SIMPLE)Computed by the programMeasured

Rule

Auto (see the table). A typed value is used as typed in both phases. It gets a warning when it does not suit the solver: below 0.9 with SIMPLEC (SIMPLEC's headroom is not used), $\alpha_p + \alpha_U > 1.1$ with plain SIMPLE, and any typed value with SIMPLEC on the compressible solver.

Source

Validation on the two-inlet test muffler, 600 iterations on the same mesh; pressure initial residual at the end (p/U/k/ω):

plain SIMPLE 0.3/0.7/0.7/0.7: 7.1e-2
plain SIMPLE 0.3/0.5/0.5/0.5: 2.3e-2
plain SIMPLE 0.3/0.7/0.3/0.3: 5.3e-2
plain SIMPLE 0.3/0.5/0.3/0.3: 4.2e-2
SIMPLEC 1/0.9/0.9/0.9: 1.1e-2, with the steadiest pressure drop (0.5 % scatter)
SIMPLEC 1/0.7/0.7/0.7: 1.6e-2

The pressure drop stayed between 1918 and 1951 Pa in every case. SIMPLEC is the default on this basis.

Compressible solver, case V1 (a 2.0 m multi-chamber muffler with a perforated pipe and three absorptive packings, 691k cells): with SIMPLEC, p 1.0 and ρ 1.0, the compressible solver aborted within 2-8 iterations at 0.5 and at 2.5 kg/s, also from a field that plain SIMPLE had already settled; plain SIMPLE (p 0.3, ρ 0.1) ran. This is the basis of the compressible auto set. The transonic set is that of OpenFOAM's internal hot-gas tutorial (the compressible solver/squareBend). The sum rule $\alpha_p + \alpha_U \approx 1$ for SIMPLE is a common practical guideline; the 1.1 warning threshold is the program's choice.

Valid range 0 < α ≤ 1

Values

$0 < \alpha_p \le 1$ . Smaller values are more robust and converge more slowly.

SIMPLEC on the compressible solver

A typed pressure relaxation applies SIMPLEC to the compressible solver as switched; with $\alpha_p = 1$ it aborted on a real muffler.

Pseudo-transient solver

the transient compressible solver with local time stepping relaxes nothing (all factors 1); its stability comes from the local Courant number (Local Courant number (pseudo-transient)).

Theory

Equations

Default and rule

Valid range 0 < α ≤ 1

Related 7

References 2